scholarly journals Generalized Fractional Inequalities of the Hermite–Hadamard Type for Convex Stochastic Processes

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
McSylvester Ejighikeme Omaba ◽  
Eze R. Nwaeze
Keyword(s):  
Stochastic Process ◽  
Integral Operator ◽  
Stochastic Processes ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Convex Stochastic Processes ◽  
Katugampola Fractional Integral ◽  
Convex Stochastic Process

AbstractA generalization of the Hermite–Hadamard (HH) inequality for a positive convex stochastic process, by means of a newly proposed fractional integral operator, is hereby established. Results involving the Riemann– Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl and Liouville fractional integrals are deduced as particular cases of our main result. In addition, we also apply some known HH results to obtain some estimates for the expectations of integrals of convex and p-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral, Revista Integración, temas de matemáticas 36 (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction.

scholarly journals Hermite-Hadamard type inequalities for p-convex stochastic processes

2019 ◽  
Vol 9 (2) ◽  
pp. 148-153 ◽  
Author(s):  
Nurgul Okur ◽  
Imdat Işcan ◽  
Emine Yuksek Dizdar
Keyword(s):  
Stochastic Process ◽  
Mathematical Programming ◽  
Stochastic Processes ◽  
Mean Square ◽  
Important Place ◽  
Mathematical Programming Problems ◽  
Square Integrability ◽  
Convex Stochastic Processes ◽  
Convex Stochastic Process

In this study are investigated p-convex stochastic processes which are extensions of convex stochastic processes. A suitable example is also given for this process. In addition, in this case a p-convex stochastic process is increasing or decreasing, the relation with convexity is revealed. The concept of inequality as convexity has an important place in literature, since it provides a broader setting to study the optimization and mathematical programming problems. Therefore, Hermite-Hadamard type inequalities for p-convex stochastic processes and some boundaries for these inequalities are obtained in present study. It is used the concept of mean-square integrability for stochastic processes to obtain the above mentioned results.

scholarly journals Certain new weighted estimates proposing generalized proportional fractional operator in another sense

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Thabet Abdeljawad ◽  
Saima Rashid ◽  
A. A. El-Deeb ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Mathematical Physics ◽  
Fractional Integral Operator ◽  
Fractional Operator ◽  
Weighted Estimates ◽  
Fractional Differential ◽  
Positive Functions ◽  
Katugampola Fractional Integral ◽  
Hadamard Fractional Integral

Abstract The present work investigates the applicability and effectiveness of generalized proportional fractional integral ($\mathcal{GPFI}$ GPFI ) operator in another sense. We aim to derive novel weighted generalizations involving a family of positive functions n ($n\in \mathbb{N}$ n ∈ N ) for this recently proposed operator. As applications of this operator, we can generate notable outcomes for Riemann–Liouville ($\mathcal{RL}$ RL ) fractional, generalized $\mathcal{RL}$ RL -fractional operator, conformable fractional operator, Katugampola fractional integral operator, and Hadamard fractional integral operator by changing the domain. The proposed strategy is vivid, explicit, and it can be used to derive new solutions for various fractional differential equations applied in mathematical physics. Certain remarkable consequences of the main theorems are also figured.

scholarly journals A basic study of a fractional integral operator with extended Mittag-Leffler kernel

2021 ◽  
Vol 6 (11) ◽  
pp. 12757-12770
Author(s):  
Gauhar Rahman ◽  
◽  
Iyad Suwan ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad ◽  
...  
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Basic Study ◽  
Basic Properties

<abstract><p>In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.</p></abstract>

scholarly journals Chebyshev type inequalities involving generalized Katugampola fractional integral operators

2019 ◽  
Vol 50 (4) ◽  
pp. 381-390 ◽  
Author(s):  
Erhan Set ◽  
Junesang Choi ◽  
\.{I}lker Mumcu
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integral Operator ◽  
Research Subjects ◽  
Fractional Integral Operators ◽  
Katugampola Fractional Integral

A number of Chebyshev type inequalities involving various fractional integral operators have, recently, been presented.Here, motivated essentially by the earlier works and their applications in diverse research subjects, we aim to establish several Chebyshev type inequalities involving generalized Katugampola fractional integral operator. Relevant connections of the results presented here with those (known and new) involving relatively simpler and familiar fractional integral operators are also pointed out.

scholarly journals Two-weight norm inequalities for the rough fractional integrals

2001 ◽  
Vol 25 (8) ◽  
pp. 517-524 ◽  
Author(s):  
Yong Ding ◽  
Chin-Cheng Lin
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Weight Functions ◽  
Norm Inequalities ◽  
Fractional Maximal Operator

The authors give the weighted(Lp,Lq)-boundedness of the rough fractional integral operatorTΩ,αand the fractional maximal operatorMΩ,αwith two different weight functions.

scholarly journals New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szegö inequality

2018 ◽  
Vol 8 (2) ◽  
pp. 137-144 ◽  
Author(s):  
Erhan Set ◽  
Zoubir Dahmani ◽  
İlker Mumcu
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integral Operator ◽  
Research Subjects ◽  
Fractional Integral Operators ◽  
Katugampola Fractional Integral

A number of Chebyshev type inequalities involving various fractional integral operators have, recently, been presented. In this work, motivated essentially by the earlier works and their applications in diverse research subjects, we establish some new Polya-Szego inequality involving generalized Katugampola fractional integral operator and use them to prove some new fractional Chebyshev type inequalities which are extensions of the results in the paper: [On Polya-Szego and Chebyshev type inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal, 10(2) (2016)].

scholarly journals Boundedness of homogeneous fractional integrals on Lp for N/α ≤ P ≤ ∞

2002 ◽  
Vol 167 ◽  
pp. 17-33 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu
Keyword(s):  
Integral Operator ◽  
Hardy Spaces ◽  
Fractional Integral ◽  
Riesz Potential ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Potential Operator ◽  
Riesz Potential Operator ◽  
Homogeneous Fractional Integral Operator

AbstractIn this paper we study the map properties of the homogeneous fractional integral operator TΩ, α on Lp(ℝn) for n/α ≤ p ≤ ∞.We prove that if Ω satisfies some smoothness conditions on Sn−1 then TΩ, α is bounded from Ln/α(ℝn) to BMO(ℝn), and from Lp(ℝn) (n/α < p ≤ ∞) to a class of the Campanato spaces l, λ (ℝn), respectively. As the corollary of the results above, we show that when Ω satisfies some smoothness conditions on Sn−1 the homogeneous fractional integral operator TΩ, α is also bounded from Hp(ℝn) (n/(n + α) ≤ p ≤ 1) to Lq(ℝn) for 1/q = 1/p-α/n. The results are the extensions of Stein-Weiss (for p = 1) and Taibleson-Weiss’s (for n/(n + α) ≤ p < 1) results on the boundedness of the Riesz potential operator Iα on the Hardy spaces Hp(ℝn).

On the range and inversion of fractional integrals in weighted spaces

1982 ◽  
Vol 92 (1-2) ◽  
pp. 51-64 ◽  
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Integral Operator ◽  
Lebesgue Space ◽  
Inversion Formula ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Weighted Spaces ◽  
Fractional Integrals ◽  
Weight Functions ◽  
And Inversion ◽  
Unweighted Case

SynopsisThe weight functions w(x) for which the Riemann fractional integral operator Iα is bounded from the Lebesgue space Lp(wp) into Lq(wq), l/q = l/p −, have been characterized by Muckenhoupt and Wheeden. In this paper, we prove an inversion formula for Iα in the context of these weighted spaces and we also characterize the range of Iα as a subset of Lq(wq) Similar results are proved for other fractional integrals. These results may be viewed as weighted analogues of certain results of Stein and Zygmund, Herson and Heywood, Heywood, and Kober who considered the unweighted case, w(x) = l.

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